erezaT Kurimaiová Presentation of Master's thesis written ...
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Spectral properties of the damped wave equation
Tereza KurimaiováPresentation of Master's thesis written under supervision of David Krejèiøík
Czech Technical University in Prague
Aspect'19
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Contents
1 Motivation - Damped vibration of string
2 Damped wave operator on Ω ⊂ Rd with bounded damping and
Schrödinger operator
3 Results for the damped wave operator obtained using:
Lieb-Thirring inequalities
Buslaev-Faddeev-Zakharov trace formulae
Birman-Schwinger principle
4 Finite potential well
Tereza Kurimaiová (CTU in Prague) Spectral properties of damped wave equation Aspect'19 2 / 23
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Motivation - Damped vibration of string
Let Ω = (0, L), a > 0 then we have the damped wave equation with the
damping a in the form
utt + aut − uxx = 0, x ∈ (0, L), t > 0
u = u1, x ∈ (0, L), t = 0
ut = u2, x ∈ (0, L), t = 0
u = 0, x = 0, L, t > 0
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Denote
U0 :=
(u1u2
)a U(t) :=
(uut
),
then formally
d
dtU(t) =
(ututt
)=
(ut
−aut + uxx
)=
(0 I∂2
∂x2−a
)(uut
)=
(0 I∂2
∂x2−a
)U(t).
Choosing the Hilbert space
H :=(H10 (0, L)× L2(0, L), (·, ·)H
)with the inner product
(Ψ,Φ)H :=
((ψ1
ψ2
),
(φ1φ2
))H
=
∫ L
0
dψ1
dx
dφ1dx
+ ψ2φ2
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Denote
U0 :=
(u1u2
)a U(t) :=
(uut
),
then formally
d
dtU(t) =
(ututt
)=
(ut
−aut + uxx
)=
(0 I∂2
∂x2−a
)(uut
)=
(0 I∂2
∂x2−a
)U(t).
Choosing the Hilbert space
H :=(H10 (0, L)× L2(0, L), (·, ·)H
)with the inner product
(Ψ,Φ)H :=
((ψ1
ψ2
),
(φ1φ2
))H
=
∫ L
0
dψ1
dx
dφ1dx
+ ψ2φ2
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A :=
(0 Id2
dx2−a
), Dom(A) :=
(H2(0, L) ∩ H1
0 (0, L))× H1
0 (0, L).
We thus obtain an evolution problem
d
dtU(t) = AU(t), U(0) = U0
for A being densely dened, closed, unbounded, non-self-adjoint and
generating a C0-semigroup eAt .
There exists ω ∈ R, ‖eAt‖ ≤ eωt .
We dene ω0 as the smallest such ω. For the string it holds
ω0 = ωσ(A) := supRλ : λ ∈ σ(A),
Cox, S., and E. Zuazua. \The Rate at Which Energy Decays in a
Damped String." Communications in Partial Dierential Equations,
vol. 19, no. 1-2, 1994, pp. 213243.
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A :=
(0 Id2
dx2−a
), Dom(A) :=
(H2(0, L) ∩ H1
0 (0, L))× H1
0 (0, L).
We thus obtain an evolution problem
d
dtU(t) = AU(t), U(0) = U0
for A being densely dened, closed, unbounded, non-self-adjoint and
generating a C0-semigroup eAt .
There exists ω ∈ R, ‖eAt‖ ≤ eωt .
We dene ω0 as the smallest such ω. For the string it holds
ω0 = ωσ(A) := supRλ : λ ∈ σ(A),
Cox, S., and E. Zuazua. \The Rate at Which Energy Decays in a
Damped String." Communications in Partial Dierential Equations,
vol. 19, no. 1-2, 1994, pp. 213243.
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A :=
(0 Id2
dx2−a
), Dom(A) :=
(H2(0, L) ∩ H1
0 (0, L))× H1
0 (0, L).
We thus obtain an evolution problem
d
dtU(t) = AU(t), U(0) = U0
for A being densely dened, closed, unbounded, non-self-adjoint and
generating a C0-semigroup eAt .
There exists ω ∈ R, ‖eAt‖ ≤ eωt .
We dene ω0 as the smallest such ω. For the string it holds
ω0 = ωσ(A) := supRλ : λ ∈ σ(A),
Cox, S., and E. Zuazua. \The Rate at Which Energy Decays in a
Damped String." Communications in Partial Dierential Equations,
vol. 19, no. 1-2, 1994, pp. 213243.
Tereza Kurimaiová (CTU in Prague) Spectral properties of damped wave equation Aspect'19 5 / 23
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A :=
(0 Id2
dx2−a
), Dom(A) :=
(H2(0, L) ∩ H1
0 (0, L))× H1
0 (0, L).
We thus obtain an evolution problem
d
dtU(t) = AU(t), U(0) = U0
for A being densely dened, closed, unbounded, non-self-adjoint and
generating a C0-semigroup eAt .
There exists ω ∈ R, ‖eAt‖ ≤ eωt .
We dene ω0 as the smallest such ω. For the string it holds
ω0 = ωσ(A) := supRλ : λ ∈ σ(A),
Cox, S., and E. Zuazua. \The Rate at Which Energy Decays in a
Damped String." Communications in Partial Dierential Equations,
vol. 19, no. 1-2, 1994, pp. 213243.
Tereza Kurimaiová (CTU in Prague) Spectral properties of damped wave equation Aspect'19 5 / 23
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Finding optimal damping
Let 0 6= Ψ =
(ψ1
ψ2
)∈ Dom(A), AΨ = λΨ:
ψ2 = λψ1,d2
dx2ψ1 − aψ2 = λψ2, ψ1(0) = ψ1(L) = ψ2(0) = ψ2(L) = 0.
Thus
− d2
dx2ψ1 = (−λa− λ2)ψ1, ψ1(0) = ψ1(L) = 0.
We obtain
λa + λ2 = −(nπ
L
)2,
and nally
σ(A) =
1
2
(−a±
√a2 − 4
(nπL
)2)+∞
n=1
.
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Finding optimal damping
Let 0 6= Ψ =
(ψ1
ψ2
)∈ Dom(A), AΨ = λΨ:
ψ2 = λψ1,d2
dx2ψ1 − aψ2 = λψ2, ψ1(0) = ψ1(L) = ψ2(0) = ψ2(L) = 0.
Thus
− d2
dx2ψ1 = (−λa− λ2)ψ1, ψ1(0) = ψ1(L) = 0.
We obtain
λa + λ2 = −(nπ
L
)2,
and nally
σ(A) =
1
2
(−a±
√a2 − 4
(nπL
)2)+∞
n=1
.
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Finding optimal damping
Let 0 6= Ψ =
(ψ1
ψ2
)∈ Dom(A), AΨ = λΨ:
ψ2 = λψ1,d2
dx2ψ1 − aψ2 = λψ2, ψ1(0) = ψ1(L) = ψ2(0) = ψ2(L) = 0.
Thus
− d2
dx2ψ1 = (−λa− λ2)ψ1, ψ1(0) = ψ1(L) = 0.
We obtain
λa + λ2 = −(nπ
L
)2,
and nally
σ(A) =
1
2
(−a±
√a2 − 4
(nπL
)2)+∞
n=1
.
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Finding optimal damping
Let 0 6= Ψ =
(ψ1
ψ2
)∈ Dom(A), AΨ = λΨ:
ψ2 = λψ1,d2
dx2ψ1 − aψ2 = λψ2, ψ1(0) = ψ1(L) = ψ2(0) = ψ2(L) = 0.
Thus
− d2
dx2ψ1 = (−λa− λ2)ψ1, ψ1(0) = ψ1(L) = 0.
We obtain
λa + λ2 = −(nπ
L
)2,
and nally
σ(A) =
1
2
(−a±
√a2 − 4
(nπL
)2)+∞
n=1
.
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Finding optimal damping
Let 0 6= Ψ =
(ψ1
ψ2
)∈ Dom(A), AΨ = λΨ:
ψ2 = λψ1,d2
dx2ψ1 − aψ2 = λψ2, ψ1(0) = ψ1(L) = ψ2(0) = ψ2(L) = 0.
Thus
− d2
dx2ψ1 = (−λa− λ2)ψ1, ψ1(0) = ψ1(L) = 0.
We obtain
λa + λ2 = −(nπ
L
)2,
and nally
σ(A) =
1
2
(−a±
√a2 − 4
(nπL
)2)+∞
n=1
.
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ω0(A) = supRλ : λ ∈ σ(A) =
−a
2, a ≤ 2π
L
−a
2+
1
2
√a2 − 4
(πL
)2, a >
2π
L
mina≥0
ω0(A) = −πL
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ω0(A) = supRλ : λ ∈ σ(A) =
−a
2, a ≤ 2π
L
−a
2+
1
2
√a2 − 4
(πL
)2, a >
2π
L
mina≥0
ω0(A) = −πL
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ω0(A) = supRλ : λ ∈ σ(A) =
−a
2, a ≤ 2π
L
−a
2+
1
2
√a2 − 4
(πL
)2, a >
2π
L
mina≥0
ω0(A) = −πL
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1 Motivation - Damped vibration of string
2 Damped wave operator on Ω ⊂ Rd with bounded damping and
Schrödinger operator
3 Results for the damped wave operator obtained using:
Lieb-Thirring inequalities
Buslaev-Faddeev-Zakharov trace formulae
Birman-Schwinger principle
4 Finite potential well
Tereza Kurimaiová (CTU in Prague) Spectral properties of damped wave equation Aspect'19 8 / 23
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Damped wave operator on Ω ⊂ Rd with bounded damping
Let Ω ⊂ Rd and a ∈ L∞(Ω). We choose
H :=(H10 (Ω)× L2(Ω), (·, ·)H
)where
(Ψ,Φ)H :=
((ψ1
ψ2
),
(φ1φ2
))H
=
∫Ω∇ψ1∇φ1 + ψ1φ1 + ψ2φ2
and dene the damped wave operator
A :=
(0 I∆ −a
), Dom(A) := Dom(−∆)× H1
0 (Ω)
Tereza Kurimaiová (CTU in Prague) Spectral properties of damped wave equation Aspect'19 9 / 23
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Damped wave operator on Ω ⊂ Rd with bounded damping
Let Ω ⊂ Rd and a ∈ L∞(Ω). We choose
H :=(H10 (Ω)× L2(Ω), (·, ·)H
)where
(Ψ,Φ)H :=
((ψ1
ψ2
),
(φ1φ2
))H
=
∫Ω∇ψ1∇φ1 + ψ1φ1 + ψ2φ2
and dene the damped wave operator
A :=
(0 I∆ −a
), Dom(A) := Dom(−∆)× H1
0 (Ω)
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A is again densely dened, closed, unbounded, non-self-adjoint and
generates a C0-semigroup eAt
If the damping a changes sign, then there lies a positive point in the
spectrum and thus there exists an unstable solution:
Freitas, P., and Krejèiøík D. \Instability Results for the Damped Wave
Equation in Unbounded Domains." Journal of Dierential Equations,
vol. 211, no. 1, 2005, pp. 168186
Our goal: better localization of the spectrum
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A is again densely dened, closed, unbounded, non-self-adjoint and
generates a C0-semigroup eAt
If the damping a changes sign, then there lies a positive point in the
spectrum and thus there exists an unstable solution:
Freitas, P., and Krejèiøík D. \Instability Results for the Damped Wave
Equation in Unbounded Domains." Journal of Dierential Equations,
vol. 211, no. 1, 2005, pp. 168186
Our goal: better localization of the spectrum
Tereza Kurimaiová (CTU in Prague) Spectral properties of damped wave equation Aspect'19 10 / 23
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A is again densely dened, closed, unbounded, non-self-adjoint and
generates a C0-semigroup eAt
If the damping a changes sign, then there lies a positive point in the
spectrum and thus there exists an unstable solution:
Freitas, P., and Krejèiøík D. \Instability Results for the Damped Wave
Equation in Unbounded Domains." Journal of Dierential Equations,
vol. 211, no. 1, 2005, pp. 168186
Our goal: better localization of the spectrum
Tereza Kurimaiová (CTU in Prague) Spectral properties of damped wave equation Aspect'19 10 / 23
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Schrödinger operator
On L2(Ω) we dene
Sµψ := −∆ψ + µVψ, Dom(Sµ) := ψ ∈ H10 (Ω) : ∆ψ ∈ L2(Ω)
where V ∈ L∞(Ω), V −−−−−→|x |→+∞
0 and µ ∈ R
For α > 0 and a ≡ αV we have
−(µα
)2∈ σp(Sµ)⇐⇒ −∆ψ + µVψ = −
(µα
)2ψ ⇐⇒
⇐⇒ Aα(ψµαψ
)=µ
α
(ψµαψ
)⇐⇒ µ
α∈ σp(Aα)
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Schrödinger operator
On L2(Ω) we dene
Sµψ := −∆ψ + µVψ, Dom(Sµ) := ψ ∈ H10 (Ω) : ∆ψ ∈ L2(Ω)
where V ∈ L∞(Ω), V −−−−−→|x |→+∞
0 and µ ∈ R
For α > 0 and a ≡ αV we have
−(µα
)2∈ σp(Sµ)⇐⇒ −∆ψ + µVψ = −
(µα
)2ψ ⇐⇒
⇐⇒ Aα(ψµαψ
)=µ
α
(ψµαψ
)⇐⇒ µ
α∈ σp(Aα)
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1 Motivation - Damped vibration of string
2 Damped wave operator on Ω ⊂ Rd with bounded damping and
Schrödinger operator
3 Results for the damped wave operator obtained using:
Lieb-Thirring inequalities
Buslaev-Faddeev-Zakharov trace formulae
Birman-Schwinger principle
4 Finite potential well
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Let Ω = Rd . For the negative point spectrum of the Schrödinger operator
there exists the Lieb-Thirring inequalities
Nµ∑n=1
|λn(µ)|γ ≤ Lγ,d
∫Rd
(µV )γ+ d
2−
Theorem (1)
Let A be the damped wave operator with damping V . If V∓ ∈ Ld(Rd) and∫Rd
V d∓ <
1
L d2,d
,
then A has no positive, respectively negative eigenvalues.
Theorem (2)
Let A be the damped wave operator with damping V . Let µ be its
positive, respectively negative eigenvalue and V∓ ∈ Lγ+ d2 (Rd). Then
|µ|γ−d2 ≤ Lγ,d
∫Rd
Vγ+ d
2∓ .
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Let Ω = Rd . For the negative point spectrum of the Schrödinger operator
there exists the Lieb-Thirring inequalities
Nµ∑n=1
|λn(µ)|γ ≤ Lγ,d
∫Rd
(µV )γ+ d
2−
Theorem (1)
Let A be the damped wave operator with damping V . If V∓ ∈ Ld(Rd) and∫Rd
V d∓ <
1
L d2,d
,
then A has no positive, respectively negative eigenvalues.
Theorem (2)
Let A be the damped wave operator with damping V . Let µ be its
positive, respectively negative eigenvalue and V∓ ∈ Lγ+ d2 (Rd). Then
|µ|γ−d2 ≤ Lγ,d
∫Rd
Vγ+ d
2∓ .
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Let Ω = Rd . For the negative point spectrum of the Schrödinger operator
there exists the Lieb-Thirring inequalities
Nµ∑n=1
|λn(µ)|γ ≤ Lγ,d
∫Rd
(µV )γ+ d
2−
Theorem (1)
Let A be the damped wave operator with damping V . If V∓ ∈ Ld(Rd) and∫Rd
V d∓ <
1
L d2,d
,
then A has no positive, respectively negative eigenvalues.
Theorem (2)
Let A be the damped wave operator with damping V . Let µ be its
positive, respectively negative eigenvalue and V∓ ∈ Lγ+ d2 (Rd). Then
|µ|γ−d2 ≤ Lγ,d
∫Rd
Vγ+ d
2∓ .
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1 Motivation - Damped vibration of string
2 Damped wave operator on Ω ⊂ Rd with bounded damping and
Schrödinger operator
3 Results for the damped wave operator obtained using:
Lieb-Thirring inequalities
Buslaev-Faddeev-Zakharov trace formulae
Birman-Schwinger principle
4 Finite potential well
Tereza Kurimaiová (CTU in Prague) Spectral properties of damped wave equation Aspect'19 14 / 23
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Let Ω = R. For the negative point spectrum of the Schrödinger operator
we have an upper bound using the Buslaev-Faddeev-Zakharov trace
formulaeNµ∑n=1
|λn(µ)|12 ≥ −µ
4
∫RV
Zakharov, V. E., and L. D. Faddeev. \KortewegDe Vries Equation: A
Completely Integrable Hamiltonian System." Fifty Years of Mathematical
Physics, 2016, pp. 277284.
Theorem (3)
Let A be the damped wave operator with damping V ∈ L1(R, |x |dx). Letµ be its real eigenvalue. If µ > 0 and
∫R V < −4 or µ < 0 and
∫R V > 4
then
|µ| ≥(∫
R|V (x)||x | dx
)−1.
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Let Ω = R. For the negative point spectrum of the Schrödinger operator
we have an upper bound using the Buslaev-Faddeev-Zakharov trace
formulaeNµ∑n=1
|λn(µ)|12 ≥ −µ
4
∫RV
Zakharov, V. E., and L. D. Faddeev. \KortewegDe Vries Equation: A
Completely Integrable Hamiltonian System." Fifty Years of Mathematical
Physics, 2016, pp. 277284.
Theorem (3)
Let A be the damped wave operator with damping V ∈ L1(R, |x |dx). Letµ be its real eigenvalue. If µ > 0 and
∫R V < −4 or µ < 0 and
∫R V > 4
then
|µ| ≥(∫
R|V (x)||x | dx
)−1.
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Theorem (4)
Let Aα be the damped wave operator with damping αV , V ∈ L1(R, |x |dx)and it holds
∫R V ≶ 0. Then for µ ≷ 0 such that
|µ| <(∫
R |V (x)||x | dx)−1
there exists exactly one α satisfying
2
(∫RV∓
)−1≤ α ≤ ∓4
(∫RV
)−1such that µ
α is an eigenvalue Aα.
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1 Motivation - Damped vibration of string
2 Damped wave operator on Ω ⊂ Rd with bounded damping and
Schrödinger operator
3 Results for the damped wave operator obtained using:
Lieb-Thirring inequalities
Buslaev-Faddeev-Zakharov trace formulae
Birman-Schwinger principle
4 Finite potential well
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Birman-Schwinger principle
Now assume Ω = Rd and V ∈ L∞(Rd) is complex.
Then
µ ∈ σp(A) =⇒ −∆ψ + µVψ = −µ2ψ =⇒ (−∆ + µ2I )ψ = −µV 12|V |
12ψ
=⇒ V−112
(−∆ + µ2I )|V |−12 |V |
12ψ = −µ|V |
12ψ
=⇒ µ|V |12 (−∆ + µ2I )−1V 1
2|V |
12ψ = −|V |
12ψ
=⇒ Kµ|V |12ψ = −|V |
12ψ
Theorem (5, BS principle for the damped wave operator)
Let A be the damped wave operator with damping V . For µ ∈ C, Rµ 6= 0
it holds
µ ∈ σp(A)⇒ −1 ∈ σp(Kµ).
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Birman-Schwinger principle
Now assume Ω = Rd and V ∈ L∞(Rd) is complex.
Then
µ ∈ σp(A) =⇒ −∆ψ + µVψ = −µ2ψ =⇒ (−∆ + µ2I )ψ = −µV 12|V |
12ψ
=⇒ V−112
(−∆ + µ2I )|V |−12 |V |
12ψ = −µ|V |
12ψ
=⇒ µ|V |12 (−∆ + µ2I )−1V 1
2|V |
12ψ = −|V |
12ψ
=⇒ Kµ|V |12ψ = −|V |
12ψ
Theorem (5, BS principle for the damped wave operator)
Let A be the damped wave operator with damping V . For µ ∈ C, Rµ 6= 0
it holds
µ ∈ σp(A)⇒ −1 ∈ σp(Kµ).
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Birman-Schwinger principle
Now assume Ω = Rd and V ∈ L∞(Rd) is complex.
Then
µ ∈ σp(A) =⇒ −∆ψ + µVψ = −µ2ψ =⇒ (−∆ + µ2I )ψ = −µV 12|V |
12ψ
=⇒ V−112
(−∆ + µ2I )|V |−12 |V |
12ψ = −µ|V |
12ψ
=⇒ µ|V |12 (−∆ + µ2I )−1V 1
2|V |
12ψ = −|V |
12ψ
=⇒ Kµ|V |12ψ = −|V |
12ψ
Theorem (5, BS principle for the damped wave operator)
Let A be the damped wave operator with damping V . For µ ∈ C, Rµ 6= 0
it holds
µ ∈ σp(A)⇒ −1 ∈ σp(Kµ).
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Theorem (6)
Let d = 1 and A be the damped wave operator with damping V ∈ L1(R).If ‖V ‖L1 < 2 then σp(A) ⊂ µ ∈ C : Rµ ≤ 0.
Theorem (7)
Let d = 3 and A be the damped wave operator with damping
V ∈ L32 (R3). Then
σp(A) ⊂
µ ∈ C : Rµ ≤ 0 ∨ |µ| ≥ 4 3
√π
3√2 ‖V ‖
L32
.
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Theorem (6)
Let d = 1 and A be the damped wave operator with damping V ∈ L1(R).If ‖V ‖L1 < 2 then σp(A) ⊂ µ ∈ C : Rµ ≤ 0.
Theorem (7)
Let d = 3 and A be the damped wave operator with damping
V ∈ L32 (R3). Then
σp(A) ⊂
µ ∈ C : Rµ ≤ 0 ∨ |µ| ≥ 4 3
√π
3√2 ‖V ‖
L32
.
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1 Motivation - Damped vibration of string
2 Damped wave operator on Ω ⊂ Rd with bounded damping and
Schrödinger operator
3 Results for the damped wave operator obtained using:
Lieb-Thirring inequalities
Buslaev-Faddeev-Zakharov trace formulae
Birman-Schwinger principle
4 Finite potential well
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Finite potential well
V (x) =
0, x < −ba, −b < x < b
0, x > b.
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-4.0 -3.5 -3.0 -2.5 -2.0 -1.5a
0.5
1.0
1.5
2.0
2.5
3.0
3.5
μmax
LT(3/2)
LT(5/2)
Figure: The dependence of the bounds for the eigenvalues of A with b = 1
and a ∈ (−4,−1.1)
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Conclusions
We analyzed the damped wave equation with both real and complex
damping
Using the correspondence between the family of Schrödinger
operators and the damped wave operator we found various spectral
bounds for the damped wave operator which provided us with
information about the behavior of the system
Tereza Kurimaiová (CTU in Prague) Spectral properties of damped wave equation Aspect'19 23 / 23